a small soda

given 12<c<12.5
we need to find which of the following translates to the above:

A. &#9474;12 ½ - c&#9474; &#8804; ½ = 0<&#9474;12 ½ - c&#9474;<1/2
=>-12.5<|-c|<12

B. &#9474;c - 12&#9474; &#8804; ½ =0<&#9474;c - 12&#9474; &#8804; ½
=>12<|c|<12.5 correct

hence B

I think the answer is E

scoobydooby, when I tried to double check the option B, it still satisfies for c = 11.5, thus it cannot be the correct answer.

My reasoning: the absolute value function swings both ways along the number line, thus, ideally I would want to check with range/2, i.e. check with 1/4 instead of 1/2. So, reading the options, I had a hunch that E is the most probable, with next probable answer being D and so on, I checked the option E with the data provided and it satisfies the given conditions.

scoobydooby, I think I found out where you went wrong...

B: |c - 12| &#8804; ½
should convert to
-½ &#8804; (c-12) &#8804; ½
or (-½ + 12) &#8804; c &#8804; (½ + 12)

Similarly Option E &#9474;c – 12 ¼&#9474; &#8804; ¼
should be converted as:
-¼ &#8804; (c – 12 ¼) &#8804; ¼
or (-¼ + 12 ¼ )&#8804; c &#8804; (¼ + 12 ¼)
which satisfies the inequality described in the question...

for more information on the absolute value function, the wiki provides more details (link given below)

https://en.wikipedia.org/wiki/Absolute_value

thanks for the excellent explanation kapsii! get it now